Ok, it turned out the issue was caused by the firewall settings of my internet router. I did search for special firewall settings/rules required for Zwift to function correctly and found this thread:
And voila, after adjusting my firewall settings accordingly everything works fine now 
While most firewalls by default allow standard http and https ports, i.e., TCP/80 and TCP/443, the high ports required by Zwift (TCP/21587 and UDP/3022) are typically blocked. Allowing those solved the problem.
Btw, I was also wondering that I was always riding alone with no other riders visible in the game when riding in training mode. Since I am fairly new to Zwift I thought this was normal and I would only see other riders on the road when joining events. Opening those ports mentioned above did solve this issue, too. I am now seeing all the other riders who are active no matter if I am just riding in training mode or during an event.
Hope this helps.